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Difference between revisions of "Semi-group of non-linear operators"

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A one-parameter family of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841302.png" />, defined and acting on a closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841303.png" /> of a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841304.png" />, with the following properties:
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841305.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841307.png" />;
+
{{TEX|auto}}
 +
{{TEX|done}}
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841308.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s0841309.png" />;
+
A one-parameter family of operators  $  S ( t ) $,
 +
0 \leq  t < \infty $,
 +
defined and acting on a closed subset  $  G $
 +
of a [[Banach space|Banach space]]  $  X $,
 +
with the following properties:
  
3) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413010.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413011.png" /> (with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413012.png" />) is continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413014.png" />.
+
1) $  S ( t + \tau ) x = S ( t ) ( S ( \tau ) x ) $
 +
for  $  x \in C $,
 +
$  t , \tau > 0 $;
  
A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413016.png" /> is of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413017.png" /> if
+
2)  $  S ( 0 ) x = x $
 +
for any  $  x \in C $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413018.png" /></td> </tr></table>
+
3) for any  $  x \in C $,
 +
the function  $  S ( t ) x $(
 +
with values in  $  X $)
 +
is continuous with respect to  $  t $
 +
on  $  [ 0 , \infty ) $.
  
A semi-group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413019.png" /> is called a contraction semi-group.
+
A semi-group $  S ( t ) $
 +
is of type $  \omega $
 +
if
  
As in the case of semi-groups of linear operators (cf. [[Semi-group of operators|Semi-group of operators]]), one introduces the concept of the generating operator (or infinitesimal generator) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413020.png" /> of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413021.png" />:
+
$$
 +
\| S ( t ) x - S ( t ) y \|  \leq  e ^ {\omega t } \| x - y \| ,\ \
 +
x , y \in C ,\  t> 0 .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413022.png" /></td> </tr></table>
+
A semi-group of type  $  0 $
 +
is called a contraction semi-group.
  
for those elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413023.png" /> for which the limit exists. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413024.png" /> is a contraction semi-group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413025.png" /> is a dissipative operator. Recall that an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413026.png" /> on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413027.png" /> is dissipative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413028.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413030.png" />. A dissipative operator may be multi-valued, in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413031.png" /> in the definition stands for any of its values at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413032.png" />. A dissipative operator is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413034.png" />-dissipative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413035.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413037.png" /> is of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413038.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413039.png" /> is dissipative.
+
As in the case of semi-groups of linear operators (cf. [[Semi-group of operators|Semi-group of operators]]), one introduces the concept of the generating operator (or infinitesimal generator)  $  A _ {0} $
 +
of the semi-group  $  S ( t ) $:
  
The fundamental theorem on the generation of semi-groups: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413040.png" /> is a dissipative operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413041.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413042.png" /> for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413043.png" />, then there exists a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413044.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413046.png" /> such that
+
$$
 +
A _ {0} x  = \lim\limits _ {h \rightarrow 0
 +
\frac{S ( h ) x - x }{h}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413047.png" /></td> </tr></table>
+
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413048.png" /> and the convergence is uniform on any finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413049.png" />-interval. (The existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413050.png" /> can also be proved if one replaces the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413051.png" /> by the weaker condition
+
for those elements  $  x \in C $
 +
for which the limit exists. If  $  S ( t ) $
 +
is a contraction semi-group,  $  A _ {0} $
 +
is a dissipative operator. Recall that an operator  $  A $
 +
on a Banach space  $  X $
 +
is dissipative if  $  \| x - y - \lambda ( Ax - Ay ) \| \geq  \| x - y \| $
 +
for  $  x , y \in \overline{ {D ( A) }}\; $,
 +
$  \lambda > 0 $.  
 +
A dissipative operator may be multi-valued, in which case  $  Ax $
 +
in the definition stands for any of its values at  $  x $.  
 +
A dissipative operator is said to be $  m $-
 +
dissipative if $  \mathop{\rm Range} ( I - \lambda A ) = X $
 +
for  $  \lambda > 0 $.  
 +
If  $  S ( t ) $
 +
is of type  $  \omega $,
 +
then  $  A _ {0} - \omega I $
 +
is dissipative.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413052.png" /></td> </tr></table>
+
The fundamental theorem on the generation of semi-groups: If  $  A - \omega I $
 +
is a dissipative operator and  $  \mathop{\rm Range} ( I - \lambda A ) $
 +
contains  $  D ( A ) $
 +
for sufficiently small  $  \lambda > 0 $,
 +
then there exists a semi-group  $  S _ {A} ( t ) $
 +
of type  $  \omega $
 +
on  $  \overline{ {D ( A ) }}\; $
 +
such that
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413053.png" /> is the distance between sets.)
+
$$
 +
S _ {A} ( t ) x  = \lim\limits _ {n \rightarrow \infty } \
 +
\left ( I -
 +
\frac{t}{n}
 +
A \right ) ^ {-} n x ,
 +
$$
  
For any operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413054.png" /> one has a corresponding Cauchy problem
+
where  $  x \in \overline{ {D ( A ) }}\; $
 +
and the convergence is uniform on any finite  $  t $-
 +
interval. (The existence of  $  S _ {A} ( t ) $
 +
can also be proved if one replaces the condition  $  \mathop{\rm Range} ( I - \lambda A ) \supset \overline{ {D ( A ) }}\; $
 +
by the weaker condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$
 +
\lim\limits _ {\overline{ {\lambda \downarrow 0 }}\; } \
 +
\lambda  ^ {-} 1 d (  \mathop{\rm Range} ( I - \lambda A ) , x ) =  0 ,
 +
$$
  
If the problem (*) has a strong solution, i.e. if there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413056.png" /> which is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413057.png" />, absolutely continuous on any compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413058.png" />, takes values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413059.png" /> for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413060.png" />, has a strong derivative for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413061.png" />, and satisfies the relation (*), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413062.png" />. Any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413063.png" /> is a unique integral solution of the problem (*).
+
where  $  d $
 +
is the distance between sets.)
  
Under the assumptions of the fundamental theorem, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413064.png" /> is a [[Reflexive space|reflexive space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413065.png" /> is closed (cf. [[Closed operator|Closed operator]]), then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413066.png" /> yields a strong solution of the Cauchy problem (*) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413067.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413068.png" /> almost everywhere, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413069.png" /> is the set of elements of minimal norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413070.png" />. In that case the generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413071.png" /> of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413072.png" /> is densely defined: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413073.png" />. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413075.png" /> are uniformly convex, then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413076.png" /> is single-valued and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413077.png" /> there exists a right derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413078.png" />; this function is continuous from the right on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413079.png" />, and continuous at all points with the possible exception of a countable set; in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413081.png" />.
+
For any operator $  A $
 +
one has a corresponding Cauchy problem
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413082.png" /> is reflexive (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413083.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413084.png" /> is separable) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413085.png" /> is a single-valued operator and has the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413086.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413088.png" /> in the [[Weak topology|weak topology]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413089.png" /> (respectively, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413090.png" />) imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413091.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413093.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413094.png" /> is a weakly (weak-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413095.png" />) continuously-differentiable solution of the problem (*). In the non-reflexive case, examples are known where the assumptions of the fundamental theorem hold with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413096.png" /> and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413097.png" /> do not even have weak derivatives on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413098.png" /> at any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s08413099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130100.png" />.
+
$$ \tag{* }
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130101.png" /> be a continuous operator, defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130102.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130103.png" /> is dissipative. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130104.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130106.png" />, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130107.png" /> the problem (*) has a unique continuously-differentiable solution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130108.png" />, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130109.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130110.png" /> is continuous on its closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130111.png" />, then it will be the generating operator of a semi-group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130112.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130113.png" /> if only and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130114.png" /> is dissipative and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130115.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130116.png" />.
+
\frac{du}{dt}
 +
( t)  \in  Au ( t ) ,\  t > 0 ,\  u ( 0= x .
 +
$$
  
In a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130117.png" />, a contraction semi-group on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130118.png" /> may be extended to a contraction semi-group on a closed convex subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130119.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130120.png" />. Moreover, the generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130121.png" /> of the extended semi-group is defined on a set dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130122.png" />. There exists a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130123.png" />-dissipative operator such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130124.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130125.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130126.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130127.png" />-dissipative, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130128.png" /> is convex and there exists a unique contraction semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130129.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130130.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130131.png" />.
+
If the problem (*) has a strong solution, i.e. if there exists a function  $  u ( t ) $
 +
which is continuous on $  [ 0 , \infty ) $,
 +
absolutely continuous on any compact subset of $  ( 0, \infty ) $,
 +
takes values in  $  D ( A ) $
 +
for almost all  $  t > 0 $,
 +
has a strong derivative for almost all  $  t > 0 $,
 +
and satisfies the relation (*), then $  u ( t ) = S _ {A} ( t ) x $.  
 +
Any function  $  S _ {A} ( t ) x $
 +
is a unique integral solution of the problem (*).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130132.png" /> be a convex semi-continuous functional defined on a real Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130133.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130134.png" /> be its [[Subdifferential|subdifferential]]; then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130135.png" /> (for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130136.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130137.png" /> is non-empty) is dissipative. The semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130138.png" /> possesses properties similar to those of a linear analytic semi-group. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130139.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130140.png" />) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130141.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130142.png" /> is a strong solution of the Cauchy problem (*), with
+
Under the assumptions of the fundamental theorem, if  $  X $
 +
is a [[Reflexive space|reflexive space]] and $  A $
 +
is closed (cf. [[Closed operator|Closed operator]]), then the function  $  u ( t ) = S _ {A} ( t ) x $
 +
yields a strong solution of the Cauchy problem (*) for $  x \in D ( A) $,
 +
with  $  ( du / dt) ( t) \in A  ^ {0} u ( t ) $
 +
almost everywhere, where  $  A  ^ {0} z $
 +
is the set of elements of minimal norm in  $  Az $.  
 +
In that case the generating operator  $  A _ {0} $
 +
of the semi-group $  S _ {A} ( t ) $
 +
is densely defined: $  \overline{ {D ( A _ {0} ) }}\; = \overline{ {D ( A ) }}\; $.  
 +
If, moreover,  $  X $
 +
and  $  X  ^  \prime  $
 +
are uniformly convex, then the operator  $  A  ^ {0} $
 +
is single-valued and for all  $  t \geq  0 $
 +
there exists a right derivative  $  d  ^ {+} u / dt = A  ^ {0} u ( t ) $;
 +
this function is continuous from the right on  $  [ 0, \infty ) $,  
 +
and continuous at all points with the possible exception of a countable set; in this case  $  D ( A _ {0} ) = D ( A ) $
 +
and  $  A _ {0} = A  ^ {0} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130143.png" /></td> </tr></table>
+
If  $  X $
 +
is reflexive (or  $  X = Y  ^  \prime  $,
 +
where  $  Y $
 +
is separable) and  $  A $
 +
is a single-valued operator and has the property that  $  x _ {n} \rightarrow x $
 +
in  $  X $
 +
and  $  Ax _ {n} \rightarrow y $
 +
in the [[Weak topology|weak topology]]  $  \sigma ( X , X  ^  \prime  ) $(
 +
respectively, in  $  \sigma ( X , Y ) $)
 +
imply  $  y = Ax $,
 +
then  $  u ( t) \in D ( A ) $,
 +
$  t \geq  0 $,
 +
and  $  u ( t ) $
 +
is a weakly (weak- $  * $)
 +
continuously-differentiable solution of the problem (*). In the non-reflexive case, examples are known where the assumptions of the fundamental theorem hold with  $  \overline{ {D ( A) }}\; = X $
 +
and the functions  $  u ( t ) = S _ {A} ( t ) x $
 +
do not even have weak derivatives on  $  X $
 +
at any  $  x \in X $,
 +
$  t \geq  0 $.
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130144.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130145.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130146.png" /> attains its minimum, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130147.png" /> converges weakly to some minimum point as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130148.png" />.
+
Let  $  A $
 +
be a continuous operator, defined on all of  $  X $,
 +
such that  $  A - \omega I $
 +
is dissipative. Then  $  \mathop{\rm Range} ( I - \lambda A ) = X $
 +
for  $  \lambda > 0 $,  
 +
$  \lambda \omega < 1 $,
 +
and for any  $  x \in X $
 +
the problem (*) has a unique continuously-differentiable solution on  $  [ 0 , \infty ) $,
 +
given by  $  u ( t ) = S _ {A} ( t ) x $.  
 +
If $  A $
 +
is continuous on its closed domain  $  D ( A ) $,  
 +
then it will be the generating operator of a semi-group of type  $  \omega $
 +
on  $  D ( A) $
 +
if only and only if  $  A - \omega I $
 +
is dissipative and  $  \lim\limits _ {\lambda \rightarrow 0 }  \lambda  ^ {-} 1 d ( x + \lambda Ax , D ( A ) ) = 0 $
 +
for  $  x \in D ( A) $.
  
Theorems about the approximation of semi-groups play an essential role in the approximate solution of Cauchy problems. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130149.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130150.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130151.png" /> be Banach spaces; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130153.png" /> be operators defined and single-valued on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130154.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130155.png" />, respectively, satisfying the assumptions of the fundamental theorem for the same type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130156.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130157.png" /> be linear operators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130158.png" />. Then convergence of the resolvents (cf. [[Resolvent|Resolvent]]) (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130159.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130160.png" />)
+
In a [[Hilbert space|Hilbert space]]  $  H $,
 +
a contraction semi-group on a set  $  C $
 +
may be extended to a contraction semi-group on a closed convex subset  $  \widetilde{C}  $
 +
of  $  H $.  
 +
Moreover, the generating operator  $  A _ {0} $
 +
of the extended semi-group is defined on a set dense in  $  \widetilde{C}  $.  
 +
There exists a unique  $  m $-
 +
dissipative operator such that  $  \overline{ {D ( A) }}\; = C $
 +
and  $  A _ {0} = A  ^ {0} $.  
 +
If  $  A $
 +
is  $  m $-
 +
dissipative, then  $  \overline{ {D ( A) }}\; $
 +
is convex and there exists a unique contraction semi-group  $  S ( t ) = S _ {A} ( t) $
 +
on  $  \overline{ {D ( A) }}\; $
 +
such that  $  A _ {0} = A  ^ {0} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130161.png" /></td> </tr></table>
+
Let  $  \phi $
 +
be a convex semi-continuous functional defined on a real Hilbert space  $  H $
 +
and let  $  \partial  \phi $
 +
be its [[Subdifferential|subdifferential]]; then the operator  $  Ax = - \partial  \phi ( x) $(
 +
for all  $  x $
 +
such that  $  \partial  \phi ( x) $
 +
is non-empty) is dissipative. The semi-group  $  S _ {A} ( t ) $
 +
possesses properties similar to those of a linear analytic semi-group. In particular,  $  S _ {A} ( t) x \in D ( A) $(
 +
$  t> 0 $)
 +
for any  $  x \in \overline{ {D ( A) }}\; $,
 +
and  $  u ( t ) = S _ {A} ( t) x $
 +
is a strong solution of the Cauchy problem (*), with
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130162.png" /> implies convergence of the semi-groups
+
$$
 +
\left \|
 +
\frac{d  ^ {+} u }{dt}
 +
( t) \right \|  = \
 +
\| A  ^ {0} u ( t ) \|  \leq 
 +
\frac{2}{t}
 +
\
 +
\| x - v \| + 2 \| A  ^ {0} v \|
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130163.png" /></td> </tr></table>
+
for all  $  t > 0 $,
 +
$  v \in D ( A) $.
 +
If  $  \phi $
 +
attains its minimum, then  $  u ( t ) $
 +
converges weakly to some minimum point as  $  t \rightarrow \infty $.
 +
 
 +
Theorems about the approximation of semi-groups play an essential role in the approximate solution of Cauchy problems. Let  $  X $,
 +
$  X _ {n} $,
 +
$  n = 1 , 2 \dots $
 +
be Banach spaces; let  $  A $,
 +
$  A _ {n} $
 +
be operators defined and single-valued on  $  X $,
 +
$  X _ {n} $,
 +
respectively, satisfying the assumptions of the fundamental theorem for the same type  $  \omega $;  
 +
let  $  p _ {n} : X \rightarrow X _ {n} $
 +
be linear operators,  $  \| p _ {n} \| _ {X \rightarrow X _ {n}  } \leq  \textrm{ const } $.  
 +
Then convergence of the resolvents (cf. [[Resolvent|Resolvent]]) ( $  \lambda > 0 $,
 +
$  \lambda \omega < 1 $)
 +
 
 +
$$
 +
\| ( I - \lambda A _ {n} )  ^ {-} 1 p _ {n} x - p _ {n} ( I - \lambda A )  ^ {-} 1
 +
x \| _ {X _ {n}  }  \rightarrow  0
 +
$$
 +
 
 +
for  $  x \in \overline{ {D ( A) }}\; $
 +
implies convergence of the semi-groups
 +
 
 +
$$
 +
\| S _ {A _ {n}  } ( t) p _ {n} x - p _ {n} S _ {A} ( t) x \| _ {X _ {n}  }
 +
\rightarrow  0 ,\  x \in \overline{ {D ( A) }}\; ,
 +
$$
  
 
uniformly on any finite closed interval.
 
uniformly on any finite closed interval.
  
The multiplicative formulas developed by S. Lie in the finite-dimensional linear case can be generalized to the non-linear case. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130164.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130165.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130166.png" /> are single-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130167.png" />-dissipative operators on a Hilbert space and the closed convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130168.png" /> is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130170.png" />, then, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130171.png" />,
+
The multiplicative formulas developed by S. Lie in the finite-dimensional linear case can be generalized to the non-linear case. If $  A $,  
 +
$  B $
 +
and $  A + B $
 +
are single-valued $  m $-
 +
dissipative operators on a Hilbert space and the closed convex set $  C \subset  \overline{ {D ( A) }}\; \cap \overline{ {D ( B) }}\; $
 +
is invariant under $  ( I - \lambda A )  ^ {-} 1 $
 +
and $  ( I - \lambda B )  ^ {-} 1 $,  
 +
then, for any $  x \in C \cap \overline{ {D ( A) \cap D ( B) }}\; $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130172.png" /></td> <td valign="top" style="width:5%;text-align:right;">(**)</td></tr></table>
+
$$ \tag{** }
 +
S _ {A + B }  ( t) x  =  \lim\limits _ {n \rightarrow \infty } \
 +
\left [ S _ {A} \left (
  
This formula is also valid in an arbitrary Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130173.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130174.png" />, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130175.png" /> is a densely-defined <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130176.png" />-dissipative linear operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130177.png" /> is a continuous dissipative operator defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130178.png" />. In both cases
+
\frac{t}{n}
 +
\right ) S _ {B} \left (
 +
\frac{t}{n}
 +
\right ) \right ]  ^ {n} x .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130179.png" /></td> </tr></table>
+
This formula is also valid in an arbitrary Banach space  $  X $
 +
for any  $  x \in X $,
 +
provided  $  A $
 +
is a densely-defined  $  m $-
 +
dissipative linear operator and  $  B $
 +
is a continuous dissipative operator defined on all of  $  X $.  
 +
In both cases
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130180.png" /></td> </tr></table>
+
$$
 +
S _ {A + B }  ( t) x  = \lim\limits \
 +
\left [ \left ( I -
 +
\frac{t}{n}
 +
B \right )  ^ {-} 1
 +
\left ( I -
 +
\frac{t}{n}
 +
A \right )  ^ {-} 1 \right ]  ^ {n} x ,
 +
$$
  
Examples of non-linear differential operators satisfying the conditions of the fundamental theorem on the generation of semi-groups are given below. In each case only the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130181.png" /> and the boundary conditions are indicated, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130182.png" /> is not described. In all examples, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130183.png" /> is a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130184.png" /> with smooth boundary; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130185.png" /> are multi-valued maximal monotone mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130186.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130187.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130188.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130189.png" /> is a continuous strictly-increasing function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130190.png" />.
+
$$
 +
x  \in  \overline{ {D ( A) \cap D ( B) }}\; .
 +
$$
 +
 
 +
Examples of non-linear differential operators satisfying the conditions of the fundamental theorem on the generation of semi-groups are given below. In each case only the space $  X $
 +
and the boundary conditions are indicated, while $  D ( A) $
 +
is not described. In all examples, $  \Omega $
 +
is a bounded domain in $  \mathbf R  ^ {n} $
 +
with smooth boundary; $  \beta , \gamma $
 +
are multi-valued maximal monotone mappings $  \mathbf R \rightarrow \mathbf R $,
 +
$  \beta ( 0) \ni 0 $,  
 +
$  \gamma ( 0) \ni 0 $;  
 +
and $  \psi : \mathbf R \rightarrow \mathbf R $
 +
is a continuous strictly-increasing function, $  \psi ( 0) = 0 $.
  
 
==Example 1.==
 
==Example 1.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130191.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130192.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130193.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130194.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130195.png" />.
+
$  X = L _ {p} ( \Omega ) $,  
 +
$  1 \leq  p \leq  \infty $,  
 +
$  Au = \Delta u - \beta ( u) $,  
 +
$  - \partial  u / \partial  n \in \gamma ( u) $
 +
on $  \Gamma $.
  
 
==Example 2.==
 
==Example 2.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130196.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130197.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130198.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130199.png" />.
+
$  X = L _ {1} ( \Omega ) $,  
 +
$  Au = \Delta \psi ( u) $,  
 +
$  - \partial  u / \partial  n \in \gamma ( u) $
 +
on $  \Gamma $.
  
 
==Example 3.==
 
==Example 3.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130200.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130202.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130203.png" />.
+
$  X = W _ {2}  ^ {-} 1 ( \Omega ) $,  
 +
$  Au = \Delta \psi ( u) $,  
 +
$  u = 0 $
 +
on $  \Gamma $.
  
 
==Example 4.==
 
==Example 4.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130204.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130207.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130208.png" />.
+
$  X = C ( \overline \Omega \; ) $
 +
or $  X = L _  \infty  ( \Omega ) $,  
 +
$  Au = \psi ( \Delta u ) $,  
 +
$  u = 0 $
 +
on $  \Gamma $.
  
 
==Example 5.==
 
==Example 5.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130210.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130211.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130212.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130213.png" />.
+
$  X = L _ {1} ( \mathbf R  ^ {n} ) $,  
 +
$  Au = \mathop{\rm div}  f ( u) $,  
 +
where $  f \in C  ^ {1} ( \mathbf R ) $
 +
with values in $  \mathbf R  ^ {n} $,
 +
$  f ( 0) = 0 $.
  
 
==Example 6.==
 
==Example 6.==
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130214.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130215.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130216.png" /> is continuous.
+
$  X = L _  \infty  ( \mathbf R ) $,  
 +
$  Au = f ( u _ {x} ) $,  
 +
where $  f : \mathbf R \rightarrow \mathbf R $
 +
is continuous.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V. Barbu,  "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici  (1976)  (Translated from Rumanian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Brézis,  "Opérateurs maximaux monotones et semigroups de contractions dans les espaces de Hilbert" , North-Holland  (1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Brézis,  A. Pazy,  "Convergence and approximation of semigroups of nonlinear operators in Banach spaces"  ''J. Funct. Anal.'' , '''9''' :  1  (1972)  pp. 63–74</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.G. Crandall,  T.M. Liggett,  "Generation of semi-groups of nonlinear transformations on general Banach spaces"  ''Amer. J. Math.'' , '''93''' :  2  (1971)  pp. 265–298</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Y. Kobayashi,  "Difference approximation of Gauchy problems for quasi-dissipative operators and generation of nonlinear semigroups"  ''J. Math. Soc. Japan'' , '''27''' :  4  (1975)  pp. 640–665</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Y. Konishi,  "On the uniform convergence of a finite difference scheme for a nonlinear heat equation"  ''Proc. Japan. Acad.'' , '''48''' :  2  (1972)  pp. 62–66</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R.H. Martin,  "Differential equations on closed subsets of a Banach space"  ''Trans. Amer. Math. Soc.'' , '''179'''  (1973)  pp. 399–414</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  G.F. Webb,  "Continuous nonlinear perturbations of linear accretive operators in Banach spaces"  ''J. Funct. Anal.'' , '''10''' :  2  (1972)  pp. 191–203</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M.I. [M.I. Khazan] Hazan,  "Nonlinear evolution equations in locally convex spaces"  ''Soviet Math. Dokl.'' , '''14''' :  5  (1973)  pp. 1608–1614  ''Dokl. Akad. Nauk SSSR'' , '''212''' :  6  (1973)  pp. 1309–1312</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.I. [M.I. Khazan] Hazan,  "Differentiability of nonlinear semigroups and the classical solvability of nonlinear boundary value problems for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130217.png" />"  ''Soviet Math. Dokl.'' , '''17''' :  3  (1976)  pp. 839–843  ''Dokl. Akad. Nauk SSSR'' , '''228''' :  4  (1976)  pp. 805–808</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V. Barbu,  "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici  (1976)  (Translated from Rumanian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Brézis,  "Opérateurs maximaux monotones et semigroups de contractions dans les espaces de Hilbert" , North-Holland  (1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Brézis,  A. Pazy,  "Convergence and approximation of semigroups of nonlinear operators in Banach spaces"  ''J. Funct. Anal.'' , '''9''' :  1  (1972)  pp. 63–74</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.G. Crandall,  T.M. Liggett,  "Generation of semi-groups of nonlinear transformations on general Banach spaces"  ''Amer. J. Math.'' , '''93''' :  2  (1971)  pp. 265–298</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Y. Kobayashi,  "Difference approximation of Gauchy problems for quasi-dissipative operators and generation of nonlinear semigroups"  ''J. Math. Soc. Japan'' , '''27''' :  4  (1975)  pp. 640–665</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Y. Konishi,  "On the uniform convergence of a finite difference scheme for a nonlinear heat equation"  ''Proc. Japan. Acad.'' , '''48''' :  2  (1972)  pp. 62–66</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R.H. Martin,  "Differential equations on closed subsets of a Banach space"  ''Trans. Amer. Math. Soc.'' , '''179'''  (1973)  pp. 399–414</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  G.F. Webb,  "Continuous nonlinear perturbations of linear accretive operators in Banach spaces"  ''J. Funct. Anal.'' , '''10''' :  2  (1972)  pp. 191–203</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M.I. [M.I. Khazan] Hazan,  "Nonlinear evolution equations in locally convex spaces"  ''Soviet Math. Dokl.'' , '''14''' :  5  (1973)  pp. 1608–1614  ''Dokl. Akad. Nauk SSSR'' , '''212''' :  6  (1973)  pp. 1309–1312</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.I. [M.I. Khazan] Hazan,  "Differentiability of nonlinear semigroups and the classical solvability of nonlinear boundary value problems for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130217.png" />"  ''Soviet Math. Dokl.'' , '''17''' :  3  (1976)  pp. 839–843  ''Dokl. Akad. Nauk SSSR'' , '''228''' :  4  (1976)  pp. 805–808</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Line 99: Line 346:
 
The formula (**) above, especially in the form
 
The formula (**) above, especially in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130218.png" /></td> </tr></table>
+
$$
 +
e ^ {it ( A+ B) }  = {s - \lim\limits } _ {n \rightarrow \infty } \
 +
( e ^ {itA/n } e ^ {itB/n } )  ^ {n} ,
 +
$$
  
which holds, e.g., when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130219.png" /> are self-adjoint operators on a separable Hilbert space so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130220.png" />, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084130/s084130221.png" />, is self-adjoint, is known as the Trotter product formula, [[#References|[a5]]], [[#References|[a4]]].
+
which holds, e.g., when $  A, B $
 +
are self-adjoint operators on a separable Hilbert space so that $  A+ B $,  
 +
defined on $  D ( A) \cap D ( B) $,  
 +
is self-adjoint, is known as the Trotter product formula, [[#References|[a5]]], [[#References|[a4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Ph. Clément,  H.J.A.M. Heijmans,  S. Angenent,  C.J. van Duijn,  B. de Pagter,  "One-parameter semigroups" , ''CWI Monographs'' , '''5''' , North-Holland  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Pazy,  "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.H. Martin,  "Nonlinear operators and differential equations in Banach spaces" , Wiley  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Simon,  "Functional integration and quantum physics" , Acad. Press  (1979)  pp. 4–6</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Trotter,  "On the product of semigroups of operators"  ''Proc. Amer. Math. Soc.'' , '''10'''  (1959)  pp. 545–551</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Ph. Clément,  H.J.A.M. Heijmans,  S. Angenent,  C.J. van Duijn,  B. de Pagter,  "One-parameter semigroups" , ''CWI Monographs'' , '''5''' , North-Holland  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Pazy,  "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.H. Martin,  "Nonlinear operators and differential equations in Banach spaces" , Wiley  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Simon,  "Functional integration and quantum physics" , Acad. Press  (1979)  pp. 4–6</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Trotter,  "On the product of semigroups of operators"  ''Proc. Amer. Math. Soc.'' , '''10'''  (1959)  pp. 545–551</TD></TR></table>

Latest revision as of 08:13, 6 June 2020


A one-parameter family of operators $ S ( t ) $, $ 0 \leq t < \infty $, defined and acting on a closed subset $ G $ of a Banach space $ X $, with the following properties:

1) $ S ( t + \tau ) x = S ( t ) ( S ( \tau ) x ) $ for $ x \in C $, $ t , \tau > 0 $;

2) $ S ( 0 ) x = x $ for any $ x \in C $;

3) for any $ x \in C $, the function $ S ( t ) x $( with values in $ X $) is continuous with respect to $ t $ on $ [ 0 , \infty ) $.

A semi-group $ S ( t ) $ is of type $ \omega $ if

$$ \| S ( t ) x - S ( t ) y \| \leq e ^ {\omega t } \| x - y \| ,\ \ x , y \in C ,\ t> 0 . $$

A semi-group of type $ 0 $ is called a contraction semi-group.

As in the case of semi-groups of linear operators (cf. Semi-group of operators), one introduces the concept of the generating operator (or infinitesimal generator) $ A _ {0} $ of the semi-group $ S ( t ) $:

$$ A _ {0} x = \lim\limits _ {h \rightarrow 0 } \frac{S ( h ) x - x }{h} $$

for those elements $ x \in C $ for which the limit exists. If $ S ( t ) $ is a contraction semi-group, $ A _ {0} $ is a dissipative operator. Recall that an operator $ A $ on a Banach space $ X $ is dissipative if $ \| x - y - \lambda ( Ax - Ay ) \| \geq \| x - y \| $ for $ x , y \in \overline{ {D ( A) }}\; $, $ \lambda > 0 $. A dissipative operator may be multi-valued, in which case $ Ax $ in the definition stands for any of its values at $ x $. A dissipative operator is said to be $ m $- dissipative if $ \mathop{\rm Range} ( I - \lambda A ) = X $ for $ \lambda > 0 $. If $ S ( t ) $ is of type $ \omega $, then $ A _ {0} - \omega I $ is dissipative.

The fundamental theorem on the generation of semi-groups: If $ A - \omega I $ is a dissipative operator and $ \mathop{\rm Range} ( I - \lambda A ) $ contains $ D ( A ) $ for sufficiently small $ \lambda > 0 $, then there exists a semi-group $ S _ {A} ( t ) $ of type $ \omega $ on $ \overline{ {D ( A ) }}\; $ such that

$$ S _ {A} ( t ) x = \lim\limits _ {n \rightarrow \infty } \ \left ( I - \frac{t}{n} A \right ) ^ {-} n x , $$

where $ x \in \overline{ {D ( A ) }}\; $ and the convergence is uniform on any finite $ t $- interval. (The existence of $ S _ {A} ( t ) $ can also be proved if one replaces the condition $ \mathop{\rm Range} ( I - \lambda A ) \supset \overline{ {D ( A ) }}\; $ by the weaker condition

$$ \lim\limits _ {\overline{ {\lambda \downarrow 0 }}\; } \ \lambda ^ {-} 1 d ( \mathop{\rm Range} ( I - \lambda A ) , x ) = 0 , $$

where $ d $ is the distance between sets.)

For any operator $ A $ one has a corresponding Cauchy problem

$$ \tag{* } \frac{du}{dt} ( t) \in Au ( t ) ,\ t > 0 ,\ u ( 0) = x . $$

If the problem (*) has a strong solution, i.e. if there exists a function $ u ( t ) $ which is continuous on $ [ 0 , \infty ) $, absolutely continuous on any compact subset of $ ( 0, \infty ) $, takes values in $ D ( A ) $ for almost all $ t > 0 $, has a strong derivative for almost all $ t > 0 $, and satisfies the relation (*), then $ u ( t ) = S _ {A} ( t ) x $. Any function $ S _ {A} ( t ) x $ is a unique integral solution of the problem (*).

Under the assumptions of the fundamental theorem, if $ X $ is a reflexive space and $ A $ is closed (cf. Closed operator), then the function $ u ( t ) = S _ {A} ( t ) x $ yields a strong solution of the Cauchy problem (*) for $ x \in D ( A) $, with $ ( du / dt) ( t) \in A ^ {0} u ( t ) $ almost everywhere, where $ A ^ {0} z $ is the set of elements of minimal norm in $ Az $. In that case the generating operator $ A _ {0} $ of the semi-group $ S _ {A} ( t ) $ is densely defined: $ \overline{ {D ( A _ {0} ) }}\; = \overline{ {D ( A ) }}\; $. If, moreover, $ X $ and $ X ^ \prime $ are uniformly convex, then the operator $ A ^ {0} $ is single-valued and for all $ t \geq 0 $ there exists a right derivative $ d ^ {+} u / dt = A ^ {0} u ( t ) $; this function is continuous from the right on $ [ 0, \infty ) $, and continuous at all points with the possible exception of a countable set; in this case $ D ( A _ {0} ) = D ( A ) $ and $ A _ {0} = A ^ {0} $.

If $ X $ is reflexive (or $ X = Y ^ \prime $, where $ Y $ is separable) and $ A $ is a single-valued operator and has the property that $ x _ {n} \rightarrow x $ in $ X $ and $ Ax _ {n} \rightarrow y $ in the weak topology $ \sigma ( X , X ^ \prime ) $( respectively, in $ \sigma ( X , Y ) $) imply $ y = Ax $, then $ u ( t) \in D ( A ) $, $ t \geq 0 $, and $ u ( t ) $ is a weakly (weak- $ * $) continuously-differentiable solution of the problem (*). In the non-reflexive case, examples are known where the assumptions of the fundamental theorem hold with $ \overline{ {D ( A) }}\; = X $ and the functions $ u ( t ) = S _ {A} ( t ) x $ do not even have weak derivatives on $ X $ at any $ x \in X $, $ t \geq 0 $.

Let $ A $ be a continuous operator, defined on all of $ X $, such that $ A - \omega I $ is dissipative. Then $ \mathop{\rm Range} ( I - \lambda A ) = X $ for $ \lambda > 0 $, $ \lambda \omega < 1 $, and for any $ x \in X $ the problem (*) has a unique continuously-differentiable solution on $ [ 0 , \infty ) $, given by $ u ( t ) = S _ {A} ( t ) x $. If $ A $ is continuous on its closed domain $ D ( A ) $, then it will be the generating operator of a semi-group of type $ \omega $ on $ D ( A) $ if only and only if $ A - \omega I $ is dissipative and $ \lim\limits _ {\lambda \rightarrow 0 } \lambda ^ {-} 1 d ( x + \lambda Ax , D ( A ) ) = 0 $ for $ x \in D ( A) $.

In a Hilbert space $ H $, a contraction semi-group on a set $ C $ may be extended to a contraction semi-group on a closed convex subset $ \widetilde{C} $ of $ H $. Moreover, the generating operator $ A _ {0} $ of the extended semi-group is defined on a set dense in $ \widetilde{C} $. There exists a unique $ m $- dissipative operator such that $ \overline{ {D ( A) }}\; = C $ and $ A _ {0} = A ^ {0} $. If $ A $ is $ m $- dissipative, then $ \overline{ {D ( A) }}\; $ is convex and there exists a unique contraction semi-group $ S ( t ) = S _ {A} ( t) $ on $ \overline{ {D ( A) }}\; $ such that $ A _ {0} = A ^ {0} $.

Let $ \phi $ be a convex semi-continuous functional defined on a real Hilbert space $ H $ and let $ \partial \phi $ be its subdifferential; then the operator $ Ax = - \partial \phi ( x) $( for all $ x $ such that $ \partial \phi ( x) $ is non-empty) is dissipative. The semi-group $ S _ {A} ( t ) $ possesses properties similar to those of a linear analytic semi-group. In particular, $ S _ {A} ( t) x \in D ( A) $( $ t> 0 $) for any $ x \in \overline{ {D ( A) }}\; $, and $ u ( t ) = S _ {A} ( t) x $ is a strong solution of the Cauchy problem (*), with

$$ \left \| \frac{d ^ {+} u }{dt} ( t) \right \| = \ \| A ^ {0} u ( t ) \| \leq \frac{2}{t} \ \| x - v \| + 2 \| A ^ {0} v \| $$

for all $ t > 0 $, $ v \in D ( A) $. If $ \phi $ attains its minimum, then $ u ( t ) $ converges weakly to some minimum point as $ t \rightarrow \infty $.

Theorems about the approximation of semi-groups play an essential role in the approximate solution of Cauchy problems. Let $ X $, $ X _ {n} $, $ n = 1 , 2 \dots $ be Banach spaces; let $ A $, $ A _ {n} $ be operators defined and single-valued on $ X $, $ X _ {n} $, respectively, satisfying the assumptions of the fundamental theorem for the same type $ \omega $; let $ p _ {n} : X \rightarrow X _ {n} $ be linear operators, $ \| p _ {n} \| _ {X \rightarrow X _ {n} } \leq \textrm{ const } $. Then convergence of the resolvents (cf. Resolvent) ( $ \lambda > 0 $, $ \lambda \omega < 1 $)

$$ \| ( I - \lambda A _ {n} ) ^ {-} 1 p _ {n} x - p _ {n} ( I - \lambda A ) ^ {-} 1 x \| _ {X _ {n} } \rightarrow 0 $$

for $ x \in \overline{ {D ( A) }}\; $ implies convergence of the semi-groups

$$ \| S _ {A _ {n} } ( t) p _ {n} x - p _ {n} S _ {A} ( t) x \| _ {X _ {n} } \rightarrow 0 ,\ x \in \overline{ {D ( A) }}\; , $$

uniformly on any finite closed interval.

The multiplicative formulas developed by S. Lie in the finite-dimensional linear case can be generalized to the non-linear case. If $ A $, $ B $ and $ A + B $ are single-valued $ m $- dissipative operators on a Hilbert space and the closed convex set $ C \subset \overline{ {D ( A) }}\; \cap \overline{ {D ( B) }}\; $ is invariant under $ ( I - \lambda A ) ^ {-} 1 $ and $ ( I - \lambda B ) ^ {-} 1 $, then, for any $ x \in C \cap \overline{ {D ( A) \cap D ( B) }}\; $,

$$ \tag{** } S _ {A + B } ( t) x = \lim\limits _ {n \rightarrow \infty } \ \left [ S _ {A} \left ( \frac{t}{n} \right ) S _ {B} \left ( \frac{t}{n} \right ) \right ] ^ {n} x . $$

This formula is also valid in an arbitrary Banach space $ X $ for any $ x \in X $, provided $ A $ is a densely-defined $ m $- dissipative linear operator and $ B $ is a continuous dissipative operator defined on all of $ X $. In both cases

$$ S _ {A + B } ( t) x = \lim\limits \ \left [ \left ( I - \frac{t}{n} B \right ) ^ {-} 1 \left ( I - \frac{t}{n} A \right ) ^ {-} 1 \right ] ^ {n} x , $$

$$ x \in \overline{ {D ( A) \cap D ( B) }}\; . $$

Examples of non-linear differential operators satisfying the conditions of the fundamental theorem on the generation of semi-groups are given below. In each case only the space $ X $ and the boundary conditions are indicated, while $ D ( A) $ is not described. In all examples, $ \Omega $ is a bounded domain in $ \mathbf R ^ {n} $ with smooth boundary; $ \beta , \gamma $ are multi-valued maximal monotone mappings $ \mathbf R \rightarrow \mathbf R $, $ \beta ( 0) \ni 0 $, $ \gamma ( 0) \ni 0 $; and $ \psi : \mathbf R \rightarrow \mathbf R $ is a continuous strictly-increasing function, $ \psi ( 0) = 0 $.

Example 1.

$ X = L _ {p} ( \Omega ) $, $ 1 \leq p \leq \infty $, $ Au = \Delta u - \beta ( u) $, $ - \partial u / \partial n \in \gamma ( u) $ on $ \Gamma $.

Example 2.

$ X = L _ {1} ( \Omega ) $, $ Au = \Delta \psi ( u) $, $ - \partial u / \partial n \in \gamma ( u) $ on $ \Gamma $.

Example 3.

$ X = W _ {2} ^ {-} 1 ( \Omega ) $, $ Au = \Delta \psi ( u) $, $ u = 0 $ on $ \Gamma $.

Example 4.

$ X = C ( \overline \Omega \; ) $ or $ X = L _ \infty ( \Omega ) $, $ Au = \psi ( \Delta u ) $, $ u = 0 $ on $ \Gamma $.

Example 5.

$ X = L _ {1} ( \mathbf R ^ {n} ) $, $ Au = \mathop{\rm div} f ( u) $, where $ f \in C ^ {1} ( \mathbf R ) $ with values in $ \mathbf R ^ {n} $, $ f ( 0) = 0 $.

Example 6.

$ X = L _ \infty ( \mathbf R ) $, $ Au = f ( u _ {x} ) $, where $ f : \mathbf R \rightarrow \mathbf R $ is continuous.

References

[1] V. Barbu, "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici (1976) (Translated from Rumanian)
[2] H. Brézis, "Opérateurs maximaux monotones et semigroups de contractions dans les espaces de Hilbert" , North-Holland (1973)
[3] H. Brézis, A. Pazy, "Convergence and approximation of semigroups of nonlinear operators in Banach spaces" J. Funct. Anal. , 9 : 1 (1972) pp. 63–74
[4] M.G. Crandall, T.M. Liggett, "Generation of semi-groups of nonlinear transformations on general Banach spaces" Amer. J. Math. , 93 : 2 (1971) pp. 265–298
[5] Y. Kobayashi, "Difference approximation of Gauchy problems for quasi-dissipative operators and generation of nonlinear semigroups" J. Math. Soc. Japan , 27 : 4 (1975) pp. 640–665
[6] Y. Konishi, "On the uniform convergence of a finite difference scheme for a nonlinear heat equation" Proc. Japan. Acad. , 48 : 2 (1972) pp. 62–66
[7] R.H. Martin, "Differential equations on closed subsets of a Banach space" Trans. Amer. Math. Soc. , 179 (1973) pp. 399–414
[8] G.F. Webb, "Continuous nonlinear perturbations of linear accretive operators in Banach spaces" J. Funct. Anal. , 10 : 2 (1972) pp. 191–203
[9] M.I. [M.I. Khazan] Hazan, "Nonlinear evolution equations in locally convex spaces" Soviet Math. Dokl. , 14 : 5 (1973) pp. 1608–1614 Dokl. Akad. Nauk SSSR , 212 : 6 (1973) pp. 1309–1312
[10] M.I. [M.I. Khazan] Hazan, "Differentiability of nonlinear semigroups and the classical solvability of nonlinear boundary value problems for the equation " Soviet Math. Dokl. , 17 : 3 (1976) pp. 839–843 Dokl. Akad. Nauk SSSR , 228 : 4 (1976) pp. 805–808

Comments

See also Semi-group of operators; One-parameter semi-group.

The formula (**) above, especially in the form

$$ e ^ {it ( A+ B) } = {s - \lim\limits } _ {n \rightarrow \infty } \ ( e ^ {itA/n } e ^ {itB/n } ) ^ {n} , $$

which holds, e.g., when $ A, B $ are self-adjoint operators on a separable Hilbert space so that $ A+ B $, defined on $ D ( A) \cap D ( B) $, is self-adjoint, is known as the Trotter product formula, [a5], [a4].

References

[a1] Ph. Clément, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de Pagter, "One-parameter semigroups" , CWI Monographs , 5 , North-Holland (1987)
[a2] A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983)
[a3] R.H. Martin, "Nonlinear operators and differential equations in Banach spaces" , Wiley (1976)
[a4] B. Simon, "Functional integration and quantum physics" , Acad. Press (1979) pp. 4–6
[a5] H. Trotter, "On the product of semigroups of operators" Proc. Amer. Math. Soc. , 10 (1959) pp. 545–551
How to Cite This Entry:
Semi-group of non-linear operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-group_of_non-linear_operators&oldid=48659
This article was adapted from an original article by S.G. KreinM.I. Khazan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article